COMPANION MATRIX FOR SUPERPOSITION OF POLYNOMIALS AND ITS APPLICATION TO KNOT THEORY

Мұқаба

Дәйексөз келтіру

Толық мәтін

Ашық рұқсат Ашық рұқсат
Рұқсат жабық Рұқсат берілді
Рұқсат жабық Тек жазылушылар үшін

Аннотация

The note provides a new formula for the companion matrix of the superposition of two polynomials over a commutative ring. The results obtained are used to provide a constructive proof of Plans’ theorem for two-bridge knots, which states that the first homology group of an odd-sheeted cyclic covering of a three-dimensional sphere branched over a given knot is the direct sum of two copies of some Abelian group. A similar result is also true for the homology of even-sheeted coverings factored by the reduced homology group of a two-sheeted covering. The structure of the above mentioned Abelian groups is described through Chebyshev polynomials of the second and fourth kind.

Авторлар туралы

A. Mednykh

Sobolev Institute of Mathematics; Novosibirsk State University

Email: smedn@mail.ru
Novosibirsk, Russia

I. Mednykh

Sobolev Institute of Mathematics; Novosibirsk State University

Email: ilyamednykh@mail.ru
Novosibirsk, Russia

G. Sokolova

Sobolev Institute of Mathematics; Novosibirsk State University; Novosibirsk State Technical University

Email: g.sokolova@g.nsu.ru
Novosibirsk, Russia

Әдебиет тізімі

  1. Bini D.A., Pan V.Y. Polynomial and Matrix Computations // Fundamental Algorithms. Birkhauser. Boston. MA. 1994. V. 1.
  2. Davis P.J. Circulant Matrices. New York: AMS Chelsea Publishing. 1994.
  3. Noferini V., Williams G. Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds // J. Algebra. 2021. V 587. P. 1-19. https://doi.org/10.1016/j.jalgebra.2021.07.018
  4. Plans A. Aportacion al estudio de los grupos de homologia de los recubrimientos ciclicos ramificados correspondiente a un nudo // Rev. R. Acad. Cienc. Exactas, Fis. Nat. Madr. 1953. V. 47. P. 161-193.
  5. Del Val P., Weber C. Plans’ theorem for links // Topol. Appl. 1990. V. 34. P. 247-255. https://doi.org/10.1016/0166-8641(90)90041-Y
  6. Gordon C. McA. A short proof of a theorem of Plans on the homology of the branched cyclic coverings of a knot // Bull. Amer. Math. Soc. 1971. V. 168. P. 85-87. https://doi.org/10.1090/S0002-9904-1971-12611-3
  7. Stevens W.H. On the Homology of Branched Cyclic Covers of Knots // LSU Historical Dissertations and Theses. 1996. doi: 10.31390/gradschool_disstheses.6282
  8. Vaserstein L.N., Wheland E. Commutators and Companion Matrices over Rings of Stable Rank 1 // Linear Algebra Appl. 1990. V. 142. P. 263-277. doi: 10.1016/0024-3795(90)90270-M
  9. Brand L. The companion matrix and its properties // Amer. Math. Monthly. 1964. V. 71. N. 6. P. 629-634. doi: 10.1080/00029890.1964.11992294
  10. Lancaster P., Tismenetsky M. The Theory of Matrices. Second Edition with Applications. San Diego: Academic press. 1985.
  11. Mason J.C., Handscomb D.C. Chebyshev Polynomials. Boca Raton: CRC Press, 2003.
  12. Nakanishi Y., Suketa M. Alexander polynomials of two-bridge knots // J. Austral. Math. Soc. (Series A). 1996. V. 60. P. 334-342. https://doi.org/10.1017/S1446788700037848
  13. Murasugi K. On the Alexander polynomial of the alternating knot // Osaka J. Math. 1958. V. 10. P. 181-189.
  14. Hartley R.I. On two-bridged knot polynomials // J. Austral. Math. Soc. (Series A). 1979. V. 28. P. 241-249. https://doi.org/10.1017/S1446788700015743
  15. Schubert H. Knoten mit zwei Bracken // Math. Z. 1956. V. 65. V. P. 133-170.
  16. Cattabriga A., Mulazzani M. Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups // Math. Proc. Cambridge Phil. Soc. 2003. V. 135. N. 1. P. 137-146. https://doi.org/10.1017/S0305004103006686
  17. Fox R.H. Free Differential Calculus III. Subgroups // Ann. Math. 1956. V. 64 N. 3. P. 407-419.
  18. Mednykh I.A. Homology group of branched cyclic covering over a 2-bridge knot of genus two. Preprint. 2021. arXiv:2111.04292 [math.CO].
  19. Seifert H. Uber das Geschlecht von Knoten // Math. Ann. 1934. V. 110. P. 571-592.
  20. Kutateladze S.S. Fundamentals of functional analysis. Netherlands: Springer Science and Business Media, 2013.
  21. Reidemeister K. Knotentheorie. New York: Chelsea Pub. Co., NewYork, 1948.
  22. Raymond Lickorish W.B. An Introduction to Knot Theory. New York: Springer, 1997.

Қосымша файлдар

Қосымша файлдар
Әрекет
1. JATS XML
Жүктеу

© Russian Academy of Sciences, 2025